广义货币交换模型的极限分布
Limiting distributions of generalized money exchange models
AI总结：
本文将即时交换与均匀储蓄模型的广义版本建模为离散时间相互作用粒子系统，刻画平稳分布并证明其渐近财富分布在适当标度下收敛于 Gamma 或指数分布。
摘要：
“货币交换模型”是一类基于智能体的仿真模型，用于研究财富分配与不平等如何通过个体之间的货币交换而演化。该模型的主要关注点是：在给定支配智能体之间交换的微观规则的情况下，识别在宏观层面出现的极限财富分布。在本文中，我们将即时交换模型和均匀储蓄模型（二者均为货币交换模型的类型）的广义版本表述为离散时间相互作用粒子系统，并刻画其平稳分布。此外，我们证明，在适当的标度下，两个模型的渐近财富分布均收敛于 Gamma 分布或指数分布。该极限分布取决于权重函数，该权重函数会影响每个智能体所交换硬币数量的概率分布。特别地，我们的结果为以往基于数值仿真和启发式论证的研究所预测的论断提供了数学上严格的表述与推广。
英文摘要：
The "Money Exchange Model" is a type of agent-based simulation model used to study how wealth distribution and inequality evolve through monetary exchanges between individuals. The primary focus of this model is to identify the limiting wealth distributions that emerge at the macroscopic level, given the microscopic rules governing the exchanges among agents. In this paper, we formulate generalized versions of the immediate exchange model and the uniform saving model both of which are types of money exchange models, as discrete-time interacting particle systems and characterize their stationary distributions. Furthermore, we prove that under appropriate scaling, the asymptotic wealth distribution converges to a Gamma distribution or an exponential distribution for both models. The limiting distribution depends on the weight function that affects the probability distribution of the number of coins exchanged by each agent. In particular, our results provide a mathematically rigorous formulation and generalization of the assertions previously predicted in studies based on numerical simulations and heuristic arguments.
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